We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to 16,384 CPU cores.

翻译：我们提出了一种随机方法，用于高效计算模拟次扩散型反常扩散问题的时间分数阶偏微分方程（fPDE）的解。在对fPDE进行空间离散化后，通过蒙特卡洛评估相应的Mittag-Leffler矩阵函数来求解所得到的分数阶线性方程组。这通过近似随机过程的一个合适乘性泛函的期望值来实现，该随机过程由一个马尔可夫链组成，其在每个状态中的停留时间服从Mittag-Leffler分布。由此产生的算法能够高效计算域中适当选定点的解。此外，我们还介绍了如何将该算法推广以计算完整解。针对若干大规模数值问题，我们的方法在共享内存和分布式内存系统中均展现出卓越性能，在多达16,384个CPU核上实现了近乎完美的可扩展性。